Calculating the number of undetected active SARS-CoV-2 infections from results of population-wide antigen tests

Current European research estimates the number of undetected active SARS-CoV-2 infections (dark figure) to be two- to 130-fold the number of detected cases. We revisited the population-wide antigen tests in Slovakia and South Tyrol and calculated the dark figure of active cases in the vulnerable populations and the number of undetected active cases per detected active case at the time of the population-wide tests. Our analysis follows three steps: using the sensitivities and specificities of the used antigen tests, we first calculated the number of test-positive individuals and the proportion of actual positives in those who participated in the antigen tests. We then calculated the dark figure in the total population of Slovakia and South Tyrol, respectively. Finally, we calculated the ratio of the dark figure in the vulnerable population to the number of newly detected infections through PCR tests. Per one positive PCR result, another 0.15 to 0.71 cases must be added in South Tyrol and 0.01 to 1.25 cases in Slovakia. The dark figure was in both countries lower than assumed by earlier studies.


INTRODUCTION
Slovakia and South Tyrol conducted population-wide antigen tests to identify individuals currently infected with "severe acute respiratory syndrome coronavirus type 2" (active cases), internationally abbreviated as SARS-CoV-2, and determine the number of undetected infections in the respective population. In Slovakia, the antigen tests were conducted on October 31 st and November 1 st , 2020. From a total population of 5,460,550 inhabitants, 3,625,332 (66.4%) persons were tested and 38,359 (1.1%) had a positive test result [1]. In South Tyrol, from a total population of 536,667 inhabitants, 361,781 (67.4%) persons were tested, and 3,615 (1.0%) had a positive test result. The antigen tests in South Tyrol were predominantly conducted between November 20 th and 22 nd . However, the official counting included antigen test results between November 18 th and 25 th [2]. The objective of this work was to calculate the number of undetected active cases, commonly known as "dark figure", from the results of the population-wide antigen tests. In contrast to previous European studies [3][4][5][6][7][8][9][10][11], our calculations are based on the diagnostic criteria of the antigen tests and therefore take false-positive and false-negative results into account. This leads to a more precise calculation of the dark figure.
JOURNAL of MEDICINE and LIFE

MATERIAL AND METHODS
Sensitivity and specificity of the used antigen tests: Slovakia used the antigen tests "Biocredit COVID-19 Ag" and "Standard Q COVID-19 Ag Test". In South Tyrol, the antigen tests "Standard Q COVID-19 Ag Test" and "Panbio COVID-19 Ag Rapid Test Device" were used [12,13]. For each antigen test, sensitivity and specificity were reported for various cycle threshold (Ct) values. As individuals might only hold a low viral load, we used the diagnostic criteria for Ct values of ≤33. For the Panbio test, the respective values were not available, so we used the specifications for Ct ≤34. In contrast to the other tests, there were two survey places mentioned for the Standard Q Test (Brazil and Germany, meanwhile and after our analysis, the corresponding values for Switzerland were published as well) with different values for sensitivity and specificity -we considered both places in our calculations. This resulted in nine combinations of sensitivity and specificity (in each case a point-estimate and two 95% confidence limits) per test and place -e.g., the upper limit of sensitivity combined with the upper limit of specificity, the upper limit of sensitivity combined with a point estimate of specificity etc. The sensitivity and specificity and the corresponding confidence interval for each test were at the time of the population-wide antigen tests:  [14][15][16]. It was not known in what proportion the individual tests were used in the population-wide tests. Therefore, we calculated the population-wide usage of each test, knowing that the true value of the dark figure will be in between the population-wide usage of the test with the worse and the test with the better diagnostic criteria.

The three steps of the analysis
First -The number of test-positive Tp cases is the sum of correct and false positives that can be mathematically expressed as: N T is the respective population size (here the population tested with antigen tests), P is the proportion of true positives in the population (here the share of active cases in the antigen test tested population), Sens is the sensitivity, and Spec the specificity of the respective test. This formula can be resolved for P to estimate the proportion of active cases in the respective antigen tested population with the formula below: P becomes negative if N T *(1-Spec) is higher than the number of test-positive persons. We excluded negative values from further analysis and reporting, as negative values would indicate that the tests would result in more false-positive than correct positive results.
Second -The dark figure was calculated with the formula below: D is the calculated dark figure for the specifications used in the first formula. As mentioned in the introduction, Slovakia tested 66.4% of its population with antigen tests, and 1.1% of the tested population had a positive result. South Tyrol tested 67.4% of its population with antigen tests, and 1.0% of the tested population had a positive result. We assumed P to be the same in N and N T . To obtain the dark figure for the total population, P must be multiplied with the total population N. This would, however, include individuals who were already identified with a "polymerase chain reaction" (PCR) test or who have just recovered from an infection, and are immune now. To forestall this selection bias, the responsible health authorities that conducted the antigen tests excluded all individuals N 90 from the antigen tests who had a positive PCR result within 90 days prior to the population-wide tests [13,17]. In Slovakia, N 90 was 55,327, and in South Tyrol N 90 was 15,293; these numbers were extracted from the respective official national and regional dashboards [18,19]. The total population N minus N 90 , the amount of PCR identified cases within the 90 days before the population-wide tests can be understood as the vulnerable population. This vulnerable population consists of persons who had not yet been infected with SARS-CoV-2 and persons who have had an infection but are vulnerable again for infection.
Third -To estimate the dark figure per detected active infection, we calculated the ratio of the dark figure D, in the respective N minus N 90 population, to the number of newly detected infections, identified with a PCR test, within 20 days before the population-wide antigen tests, because: "The median duration for SARS-CoV-2 carrying was 20 days (6 to 50 days) with a P25 of 16 days, and a P75 of 28 days" [20]. This means that we compared the number of PCR detected active infections with the number of usually undetected active infections identified with the population-wide antigen tests. The number of PCR detected cases was 38,867 in Slovakia and 10,917 for South Tyrol. The corresponding data were extracted from the respective official national and regional dashboards [18,19]. 1.0% of the tested population had a positive result. We assumed P to be the same in N and NT. To obtain the dark figure for the total population, P must be multiplied with the total population N. This would, however, include individuals who were already identified with a "polymerase chain reaction" (PCR) test or who have just recovered from an infection, and are immune now. To forestall this selection bias, the responsible health authorities that conducted the antigen tests excluded all

1.
First: The number of test-positive Tp cases is the sum of correct and false positives that can be mathematically expressed as: 1.0% of the tested population had a positive result. We assumed P to be the same in N and NT. To obtain the dark figure for the total population, P must be multiplied with the total population N. This would, however, include individuals who were already identified with a "polymerase chain reaction"

RESULTS
The results of our calculations are displayed in

DISCUSSION
So far, it was expected that a high number of SARS-CoV-2 infections were undetected. Previous European studies based their results on statistical modeling [5,[7][8][9]11], serological testing in study participants [3,6,10], and the application of study results from a third country to their own [4]. The authors report two to 130 undetected cases per detected case for Austria, Germany, Italy, Spain, and the United Kingdom [3,4,[6][7][8][9][10][11], estimates we cannot confirm. According to our calculations, 0.01 to 1.25 cases (Slovakia) and 0.15 to 0.71 (South Tyrol) cases per one positive PCR result must be added -the latter is in line with the retrospective analysis of the COVID-19 pandemic in Italy by Fochesato and colleagues [5]. At the same time, the antigen tests that took place in Slovakia, 21,477 PCR tests were performed additionally, and 4,165 (19.4%) of these were positive. The same was applied in South Tyrol, 21,210 PCR tests were performed, and 3,857 (18.2%) were positive. These positive rates were considerably higher, as they were event-related than the 1.1% and 1.0% reported by the antigen tests, respectively. Therefore, at least for Slovakia and South Tyrol, it can be assumed that the detection mechanisms in place worked well and may have been improved with antigen tests. Calculating the share of fatal infections from the assumed number of total infections, lethality is higher as the total number of infections is lower due to a smaller amount of undetected cases [21]. Our calculations show that the number of undetected cases is lower in Slovakia and South Tyrol than what could be expected from current European research. Evidently, this results in a higher share of fatal cases from the number of total infections and leads to fewer infections but an increased lethality.
It is a limitation that the proportions of the tests used in the two population-wide antigen tests are unknown to the scientific community -precise numbers would increase the precision of the calculations. In addition, the sensitivity and specificity of the antigen tests in the field application in Slovakia and South Tyrol are unknown. Variations of sensitivity and specificity may occur due to differences in testing procedures, personnel training, and other deviations. Moreover, we do not know the impact of the fact that willingness to participate in antigen testing may have varied between age groups; this is unclear, especially for children. Thus, the age distribution in the participants of the antigen tests may differ from the age distribution in the whole population. Finally, the assumption of P to be the same in N and N T is a limitation -although it is likely to be correct, we cannot prove nor deny it. The strength of our calculations is that there is no need for a stochastic model that needs a high dark figure to explain the observed PCR cases. Our calculations that use real-world data and real-world estimates show a decreased number of undetected cases, compared to current literature, and are in line with the observation that the SARS-CoV-2 pandemic has so far been less dynamic than influenza pandemics [22]. To better estimate the dark figure, official statistics should report the amount of false-positive cases in antigen and PCR tests. However, a gold standard assay is missing so far. As suggested, to reduce the number of false positives, test results need to be interpreted against the background of the individual pretest probability [23]. Mahase went one step further and criticized that the only differential diagnostic criteria to diagnose COVID-19 was the positivity of a PCR test. No other clinical features or symptoms were needed for the diagnosis [24]. Stang et al. recently demonstrated that a positive SARS-CoV-2 PCR result is not sufficient to be interpreted for COVID-19 diagnosis. Above this, it is demonstrated that asymptomatic individuals have a considerably higher Ct value than symptomatic persons [25]. Again, the lethality of COVID-19 increases when the diagnosis of the illness is based on a more differentiated, clinical image. The dark figure shrinks again as fewer cases are covered by a more differential diagnostic definition. Against the background of the considerable huge amount of false-positive antigen test results in Slovakia, which caused quarantine for truly uninfected persons -an ethical and legal dilemma, a positive antigen test result should always be verified with a PCR test. Quarantine should only be imposed on infectious persons.

CONCLUSION
The results of our calculations indicate that the number of undetected SARS-CoV-2 cases in European studies is frequently overestimated. Calculating the dark figure from the results of population-wide antigen tests by using the sensitivity and specificity of the antigen tests used, delivers reliable results for the true number of undetected SARS-CoV-2 infections in a population. Based on our calculations the true number of undetected SARS-CoV-2 cases is lower than what was expected so far, therefore lethality of SARS-CoV-2 is higher than what was expected by current research.